Question

Check the continuity of the function f: R^2 to R at (0,0)

f(x,y) = { 3x^2y/(x^2+y^2) if (x,y)≠(0,0)

{ 3 if (x,y)= (0,0)

Answer 1

Approaching $(0,0)$ along the line $y=x$

$$\lim\limits_{(x,y)\to(0,0)}\dfrac{3x^2y}{x^2+y^2}=\lim\limits_{(x,y)\to(0,0)}\dfrac{3x^3}{2x^2}=0$$

Since $f(0, 0)=3\not=0,$ the function $f$ is not continous at $(0, 0).$